Classification of artificial intelligence tools for civil engineering under the notion of complex fuzzy rough Frank aggregation operators

In recent days researchers have tried to handle the maximum information and use those techniques and methods in which there is no chance of data loss or loss of information is minimum. The structure like fuzzy set and complex fussy set cannot discuss the upper and lower approximations. Moreover, we can observe that a fuzzy rough set cannot discuss the second dimension and in this case, there is a chance of data loss. To cover all these issues in previous ideas, the notion of a complex fuzzy rough set in Cartesian form is the demand of the day because this structure can discuss the second dimension as well as upper and lower approximations. For this purpose, in this manuscript, we have developed the theory of complex fuzzy relation and complex fuzzy rough set in Cartesian form. Moreover, we have initiated the fundamental laws for complex fuzzy rough numbers based on Frank t-norm and t-conorm. The fundamental tools that can convert the overall input into a single output are called aggregation operators (AOs). So based on the characteristics of AOs, we have defined the notion of complex fuzzy rough Frank average and complex fuzzy rough Frank geometric AOs. The utilization of the developed theory is necessary to show the importance and validity of the delivered approach. So based on developed notions, we have defined an algorithm for this purpose along with an illustrative example. We have utilized the introduced structure for the classification of AI tools for civil engineering. Moreover, the comparative analysis of the delivered approach shows the advancement of the introduced structure as compared to existing notions.


Literature review
The notion of fuzzy set (FS) 9 provides a mathematical framework that connects the qualitative, unpredictability of human cognition with the demand for formal, computational procedures in diverse domains.FS and fuzzy logic provide a potent tool for dealing with and controlling uncertainty, ambiguity, and imprecision in a variety of applications.Information is not always exact or binary (true/false) in many real-world applications.Information that is uncertain or imprecise can be represented and used in FSs.For instance, it would be more accurate to describe the concept of "partly cloudy" as "fuzzy" rather than using the clear-cut crisp categories of "cloudy" or "not cloudy."The uncertainty in human language and perception is better captured by FSs.Traditional set theory is challenging to apply to natural language since it is frequently ambiguous and context-dependent.These linguistic words can be represented in a quantitative way using FSs.Different granularities of information can be modeled using FSs.FSs can represent a continuous range of heights that move gradually from one category to another, representing the reality that there is no sharp distinction between "tall" and "short," for instance, when defining a person's height as "tall" or "short."The notion of FS uses the membership grade (MG) whose range belongs to [0, 1].FS has many applications in civil engineering.Chameau et al. 10 utilize the idea of FS and they have proposed the potential application of FS in civil engineering.Chan et al. 11 provide a study of the application of fuzzy techniques in construction management research.Moreover, Ayyub and Haldar 12 study project scheduling using the FS concepts.Lorterapong and Moselhi 13 study the project network analysis using the idea of FS.Guan et al. 14 have used the notion of complex linear diophantine FS over AG-groupoids with application in civil engineering.Gurcanli and Mungen 15 study an occupational safety risk analysis method at construction sites using the concept of FS.Gong et al. 16 analyze the geotechnical design of earth slopes using the notion of FS.Also, Narayanamoorthy et al. 17 utilize a novel augmented Fermatean MCDM approach for the identification of renewable energy power plant locations.In many decision-making situations, we have discussed the second dimension of a variable.So we need to discuss such a kind of fuzzy structure that handles such kind of information.FS has its drawback that it can never consider the second dimension of a variable.So the notion of complex fuzzy set (CFS) has been initiated.Two kinds of attempts have been made in this regard, one by Ramot et al. 18 and a second by Tamir et al. 19 .In the first structure developed by Ramot et al. 18 , they used the range of MG belonging to the unit circle in a complex plane.But Tamir et al. 19 generalized this theory and used this range belonging to unit squares instead of the unit circle in the complex plane.CFS is a more generalized form than FS due to considering the second dimension (extra fuzzy information) of a variable.CFS enhances the range of data and there is less chance of data loss in this structure, whenever decision-makers want to take their data in the form of CFS.The literature is rich with the application of CFS in different sectors.Ma et al. 20 proposed a method for multiple periodic factor prediction problems using CFS.Bi et al. 21used the notion of CFS and proposed the concept of CF arithmetic AOs.Hu et al. 22 establish the orthogonality relation of CFS.Chen et al. 23 established the adaptive neuro-complex fuzzy inferential system and applied this system to the domain of time series forecasting.Bi et al. 24 proposed two classes of similarity measures for CFSs.Moreover, Tuncer et al. 25 developed a discrete fuzzy transform-based face image recognition method.
Rough set (RS) 26 theory is motivated by its capacity to deal with data uncertainty and imprecision, offers a systematic method for feature selection and reduction, helps decision-making processes, produces comprehensible results, and promotes knowledge discovery across a variety of fields.It has found use in a variety of domains,

The main contribution of this study
Based on the observations of the literature given in the above section, it is necessary for the literature to discuss notions that can handle the upper and lower approximation and can also discuss the second dimension in one structure.Moreover, the range of that structure must be a unit square.So, the idea of a complex fuzzy rough set in Cartesian form is in high demand right now because it can discuss both upper and lower approximations as well as the second dimension, which can be taken into account by some already existing structures like FS and CFS.So in this article, the theory of complex fuzzy relation and complex fuzzy rough set in Cartesian form has been developed.Moreover, we have initiated the fundamental laws for complex fuzzy rough numbers based on Frank t-norm and t-conorm.The fundamental tools that can convert the overall input into a single output are called AOs.Thus, we defined the concepts of complex fuzzy rough Frank average and complex fuzzy rough Frank geometric AOs.The utilization of the developed theory is necessary to show the importance and validity of the delivered approach.So based on developed notions, we have defined an algorithm for this purpose along with an illustrative example.We have utilized the introduced structure for the classification of AI tools for civil engineering.Moreover, the comparative analysis of the delivered approach shows the advancement of the introduced structure as compared to existing notions.The graphical representation of the proposed work is given in Fig. 1.

Study structure
The remainder of the paper is organized as follows.We have revised the definitions of RS, FS, CFS, and frank t-norm and t-conorm in "Preliminaries".The concept of a complex fuzzy rough set in Cartesian form and a complex fuzzy relation is covered in "Construction of complex fuzzy rough set (CFRS) in Cartesian form".Furthermore, the basic operating rules for complex fuzzy rough numbers are also discussed in "Construction of complex fuzzy rough set (CFRS) in Cartesian form".The complex fuzzy rough frank average and geometric

Preliminaries
In this section, we will review the basic definitions of RS, FS, CFS, and Frank t-norm and t-conorm.Definition 1 26 Let h be the universal set (US) and R el. represent an equivalence relation, the pair ( h , R el. ) is referred to as an approximation space.For a non-empty set A ⊂ h , if the set A can be expressed as the union of some equivalency classes then A is said to be definable.Otherwise, it is not definable.Now, if A is not definable then we can approximate the set A into definable subsets referred to as lower and upper approximations given by The pair . Then R el. is an equivalence relation on h .Now equivalence classes are given by Definition 2 8 The structure of FS has the form Here L(x) represent the MG and L : h Definition 3 19 For a US h , the notion of CFS is given by where L(x) represent the real part of the MG and L : Definition 4 8 The notion of Frank t-norm and Frank t-conorm is given as

Construction of complex fuzzy rough set (CFRS) in Cartesian form
To define the notion of a complex fuzzy rough set in Cartesian form has its importance and there is no such idea has been developed.For this purpose, the importance of complex fuzzy relations in Cartesian form cannot be denied.So in this part of the article, first of all, we have established the idea of complex fuzzy relation and then based on this relation we have delivered the idea of CFRS in Cartesian form.The overall discussion is given by: Definition 5 For any two sets P and Q , the structure R el. = {(p, q), M(p, q)|M(p, q) = S(p, q) + iT (p, q) where p ∈ P and q ∈ Q} that is the complex fuzzy subset of P × Q, where 0 ≤ S(p, q) ≤ 1, 0 ≤ T (p, q) ≤ 1 and M(p, q) : P × Q → [0, 1] + i[0, 1] , is called complex fuzzy relation (CFR) from P to Q.

Definition 6
For any arbitrary set P the structure of the form R el. = {(p, q), M(p, q)|M(p, q) = S(p, q) + iT (p, q) where P, q ∈ P} that is the complex fuzzy subset "R el." of P × P, where 0 ≤ S(p, q) ≤ 1, 0 ≤ T (p, q) ≤ 1 and M(p, q) : Definition 7 Assume that R el. be a CFR on P then (P, R el. ) define the complex fuzzy approximation space (CFAS).Now for any set A ∈ CFS(P), A = a (p) + ιb (p) then upper and lower approximations of A concerning (P, R el. ) is defined as follows where The pair R h For simplicity, we will say that Ĉom. (p).

Operational rules for CFRNs based on Frank t-norm and t-conorm
This section is devoted to proposing some elementary operational rules based on Frank t-norm and t-conorm for CFRNs.
be two CFRNs, ( c > 1 , and t ‡ > 0 is any real number.Then the operational rules based on Frank t-norm and t-conorm for CFRNs are given by (p) = (p 1 , 0.

Complex fuzzy rough Frank (CFRF) Aggregation operators (AOs)
In this section, we have to discuss the notion of AOs called CFRF AOs.Now throughout this section, ., n be a collection of CFRNs.Also, assume that . ., � w−n ) denote the weight vector (WV) such that 0 ≤ w−j ≤ 1 , and n j=1 w−j = 1.

Complex fuzzy rough Frank arithmetic AOs
This part of the article is devoted to defining the notion of CFRF arithmetic AOs.The overall discussion is given by: Definition 10 For the family of CFRNs, the CFR Frank weighted average (CFRFWA) AO is the function Ĉom.      2) is valid for n = k + 1 .Therefore, Eq. ( 2) is valid for all n.www.nature.com/scientificreports/Now we will discuss that CFRFWA AOs satisfy the following characteristics.Here

Complex fuzzy rough Frank ordered weighted average (CFRFOWA) AOs
The notion of CFROWA AO is defined here.Moreover, we will analyze the characteristics of this delivered approach.

Complex fuzzy rough Frank geometric (CFRFG) AOs
The notions of CFRFWG, CFRFOWG, and CFRFHWG AOs are discussed here.Furthermore, the properties of these notions have been discussed.

Complex fuzzy rough Frank weighted geometric (CFRFWG) AOs
Here, we have elaborated the structure of CFRFWG AOs.Also, we have discussed the properties of these ideas.

Definition 13
For the family of CFRNs, the CFRFWG operator is the function Ĉom.
n → Ĉom.given by

Complex fuzzy rough Frank-ordered weighted geometric (CFRFOWG) AOs
This idea of CFRFOWG AOs is elaborated here.Moreover, we have delivered the properties of these introduced ideas.

Complex fuzzy rough Frank hybrid weighted geometric (CFRHWG) AOs
By combining the characteristics of the CFRFWG and CFRFOWG, we have developed the structure of CFRF-HWG AOs.

MADM approach for the developed notions
To see the effective use of the developed approach, here in this part of the article, we have to discuss the multiattribute decision-making approach for the utilization of delivered notions.The MADM approach is one of the famous approaches for decision-making scenarios.In this approach, a specific algorithm based on developed theory is utilized for the selection of suitable alternatives from the set of given alternatives.Now we have to discuss that algorithm as follows: Assume the set G ter. = {G ter.−1 , G ter.−2 , G ter.−3 , . . ., G ter.−m } represent the alternatives set and there are "m" alternatives.Furthermore, the set of "n" attributes is given by S atr.= {S atr.−1 , S atr.−2 , S atr.−3 , . . ., S atr.−n }.Let � w = (� w−1 , � w−2 , � w−3 , . . ., � w−n ) be the WVs of attributes such that 0 ≤ w−j ≤ 1 , and n j=1 w−j = 1.Suppose decision analyst provide their assessment in the form of CFRNs and these values are given in the form of a matrix as where (i : 1, 2, . . ., m) and j : 1, 2, . . ., n .
Now the stepwise algorithm is given by:

MADM algorithm
We have to deliver the MADM algorithm under the environment of delivered ideas of CFRFWA and CFRFWG AOs as follows: Step Step 2. Now use the Definition 9 to obtain the score values Scr.Ĉom.−i (i = 1, 2, 3, . . ., m) for Ĉom.−i to rank the alternatives.If the score value for any two CFRNs are the same then we utilize the accuracy function to rank the alternatives.
Step 3. Order all the alternatives to select the best alternative.

Numerical example AI in civil engineering
The design, development, and maintenance of the physical and natural built environment are the focus of the engineering discipline known as civil engineering.It includes a broad range of tasks and initiatives, such as the design and construction of public works like highways, tunnels, bridges, airports, dams, buildings, and water and sewage systems.Civil engineers are in the position of assuring the sustainability, use, as well as security of these systems and structures, taking into account elements like cost-effectiveness, ecological impact, and choice of materials.They have a significant impact on how modern society is built and how its citizens' quality of life is maintained.Numerous industries, including civil engineering, have adopted artificial intelligence (AI) in important ways.Infrastructural project planning, design, construction, and management practices for civil engineers are being revolutionized by IT.In civil engineering, the following are some important applications of AI (1) Design optimization (2) Structural health monitoring (3) Construction Management (4) environmental impact assessment (5) Risk assessment (6) Cost estimation (7) Building information modeling (8) Safety Monitoring (9) Natural disaster Preparedness (10) Project collaboration.

Importance of AI in civil engineering
Here are some key points that show the importance of AI in civil engineering 1.Data analysis, design optimization, and project scheduling are just a few examples of the repetitive, timeconsuming jobs that AI can automate.As a result, civil engineers can concentrate on more challenging and original areas of their work.2. Large amounts of data from numerous sources, such as sensors and remote monitoring systems, can be processed by AI.It may analyze this data to offer perceptions, forecasts, and trends that can enhance planning, maintenance, and construction decision-making.3.By performing simulations and optimizing designs depending on different characteristics, AI-powered tools can help with the design of structures and systems, resulting in more affordable and reliable solutions.4. By examining past data and identifying prospective dangers, AI can assist in assessing the risks related to civil engineering projects.Engineers can reduce risks by making decisions based on this knowledge.5.By analyzing sensor data, artificial intelligence (AI) can be used to continuously monitor the performance and health of infrastructure, such as bridges and buildings.It can spot early indications of damage and suggest upkeep or repairs, improving safety.6. AI-powered project management solutions can help with planning, allocating resources, and scheduling to make sure that projects are finished on time and under budget.7. Drones and sensors with AI capabilities can check and remotely monitor infrastructure, eliminating the need for manual inspections in hazardous or difficult-to-reach areas.

AI tools for civil engineering
Different kinds of AI tools can be helpful for civil engineers.These are (1) Civil.AI (2) Autodesk generative design (3) Plaxis AI (4) Tekla Structural Designer.The discussion of these tools is given as follows: 1. Civil.AI An innovative newcomer, Civils.AI, is introducing an AI-powered platform that will revolutionize the dynamic field of civil engineering.This platform is a pioneer in its industry and one of the top AI tools for civil engineers, efficiently aiding activities related to infrastructural project design, construction, and maintenance.The AI-driven design tools offered by Civils.AI are among its most alluring features.Automation of essential processes including structural analysis, material selection, and cost estimation increases productivity and improves job quality.The design process is greatly simplified by these AI-powered tools, increasing output.

Autodesk
A well-known software company that specializes in creating CAD, 3D modeling, and engineering tools is Autodesk.The company's software solutions are extensively used across a range of sectors, including media, entertainment, manufacturing, engineering, and architecture.The following are some of the software applications and platforms offered by Autodesk (1) AutoCAD (2) Autodesk Maya (3) AutoCAD civil CD (4) AutoCAD LT etc. Professionals from all over the world use Autodesk's tools to design, model, simulate, and visualize a variety of projects.

Plaxis 3D
The computer programed PLAXIS does finite element analysis for deformation, stability, and water flow in the field of geotechnical engineering.The words "Plastic" and "AXI Symmetry," which refer to the geometric types www.nature.com/scientificreports/ Step 3.According to the results obtained from step 2, we can order the alternatives and find out the best alternative.
Hence ordering is as follows So G ter.−4 is the best alternative.

Utilization of CFRFWG AOs
Here we will utilize the notion of CFRFWG aggregation operator for the data given in Table 2.
Hence ordering is as follows So G ter.−4 is the best alternative.

Conclusion
Researchers these days strive to handle as much data as possible and employ techniques and approaches that minimize or eliminate the possibility of data loss.The discussion of upper and lower approximations is not possible in structures such as fuzzy sets and complex fussy sets.Furthermore, we can see that the fuzzy rough set is unable to address the second dimension and that there is a risk of data loss in this situation.The idea of a complex fuzzy rough set in Cartesian form is necessary to address all of the concerns raised in earlier proposals since it can address both upper and lower approximations as well as the second dimension.So in this paper, we have defined the notion of a complex fuzzy rough set.Since the delivered approach in this form uses the range of membership grade as a unit square instead of a unit circle in the complex plane this property makes it dominant to many existing notions.AOs are the mathematical structure that converts the overall information into a single value and they can help in many decision-making situations.So based on frank t-norm and t-conorm, we have further discussed the theory of frank AOs under the notion of a complex fuzzy rough set.For the utilization of the delivered approach, we have defined an algorithm that can help in decision-making situations.We have proposed an example for the classification of AI tools for civil engineering.Moreover, the comparative analysis of the delivered approach shows the dominance of the introduced structure.The established work is limited because whenever the decision-makers provide their assessment in the form of the complex fuzzy rough intuitionistic fuzzy rough and their generalized structure then the developed approach can never cover that kind of information because the introduced notion can never discuss the non-membership grade in upper and lower approximations with two dimensions.
Moreover, we can extend these notions to complex intuitionistic fuzzy rough sets and complex Pythagorean fuzzy rough sets.3.

Figure 1 .
Figure 1.Graphical representation of the established theory.
2, .., n be two collections of CFRNs, then CFRFOWA AOs follow the characteristics given by 1. (Idempotency): If all Ĉom.−j = Ĉom.∀j then 2 .B o u n d e d n e s s : L e t Ĉ− om.= min j 2, .., n be two collections of CFRNs, then the following properties hold for the structure of CFRFHWA AOs. 1. (Idempotency): If all Ĉom.−j = Ĉom.∀j then 2 .B o u n d e d n e s s : L e t Ĉ− om.